On Chernikov-by-nilpotent groups

TL;DR

Addresses the problem of describing groups $G$ in which, for every $g\in G$, the subgroup $\langle g^{X_k(G)}\rangle$ is Chernikov of size at most $(m,n)$. The authors extend Neumann–Wiegold type results by exploiting radicable abelian normal subgroups and a decomposition into Prüfer subgroups to control commutator structure. The main theorem proves that $\gamma_{k+1}(G)$ is Chernikov and $(k,m,n)$-bounded under the hypothesis. This work generalizes previous finite bounds (e.g., DDMP2021) to Chernikov bounds and advances the understanding of how bounded verbal conjugacy affects the lower central series, with potential impact on the theory of $BCC$-groups and related group-structure questions.

Abstract

Let $γ_k=[x_1,\dots,x_k]$ be the $k$-th lower central group-word. Given a group $G$, we write $X_k(G)$ for the set of $γ_k$-values and $γ_k(G)$ for the $k$-th term of the lower central of $G$. This paper deals with groups in which $\langle g^{X_k(G)} \rangle$ is a Chernikov group of size at most $(m,n)$ for all $g\in G$. The main result is that $γ_{k+1}(G)$ is a Chernikov group and its size is $(k,m,n)$-bounded.

Theorems & Definitions (4)

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• proof : Proof of Theorem \ref{['main']}